Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \(f(g(h(x)))\) and \(g(h(f(x)))\). $$ f(x)=x+2, g(x)=x^{2}, h(x)=\frac{x}{2-x} $$

Step-by-Step Solution

Answer

The results for \(f(g(h(x)))\) and \(g(h(f(x)))\) are \(f(g(h(x)))=\frac{x^2}{(2-x)^2}+2\) and \(g(h(f(x)))=\frac{(x+2)^2}{x^2}\) respectively.

Step 1: Calculate \(f(g(h(x)))\)

First off, start by substituting \(h(x)\) into \(g\). This will result in \(g(h(x))=g(\frac{x}{2-x})=(\frac{x}{2-x})^2=\frac{x^2}{(2-x)^2}\). Then, substitute \(g(h(x))\) into \(f\). This implies \(f(g(h(x)))=f(\frac{x^2}{(2-x)^2})=\frac{x^2}{(2-x)^2}+2\ .

Step 2: Calculate \(g(h(f(x)))\)

Starting by substituting \(f(x)\) into \(h\). This results into \(h(f(x))=h(x+2)=\frac{x+2}{2-(x+2)}=\frac{x+2}{2-x-2}=\frac{x+2}{-x}\). Now, substitute \(h(f(x))\) into \(g\), getting \(g(h(f(x)))=g(\frac{x+2}{-x})=(\frac{x+2}{-x})^2=\frac{(x+2)^2}{x^2}\ .

Key Concepts

Function Notation

Understanding function notation is essential when dealing with mathematical functions. It provides a simple way to represent the inputs and outputs of a function. In our case, we have three functions:

• \( f(x) = x + 2 \)
• \( g(x) = x^2 \)
• \( h(x) = \frac{x}{2-x} \)

These notations tell us how each input is transformed into an output. When you see something like \( f(g(h(x))) \), it means we apply \( h(x) \) first, then \( g(x) \), and finally \( f(x) \). This type of arrangement is crucial when solving more complex mathematical problems. It neatly organizes how each step proceeds, guiding us step-by-step through the solution.

Function notation also helps clarify whether operations are carried out simultaneously or sequentially. For example, in this instance, \( h(x) \) must be computed first, which then becomes the input to \( g(x) \), and so on.

Composite Functions

Composite functions involve the combination of two or more functions. They help us understand how different functions interact with each other. The notation \( f(g(h(x))) \) describes a composite function. Here, the function \( h(x) \) transforms the input, \( x \), which becomes the input for \( g(x) \), and then \( g(x) \) again becomes the input for \( f(x) \).

Composite functions follow the order of operations from 'inside out'. This means:

• Handle the innermost function first, in this case, \( h(x) \).
• Then, use the result as the input for the next outer function, such as \( g(x) \).
• Repeat this process until you solve for the outermost function, \( f(x) \) in this example.

The idea is to break down a complex problem into manageable stages, simplifying each stage before moving on to the next. These step-by-step substitutions allow us to systematically solve for any point \( x \) in functions composed together.

Algebraic Manipulation

Algebraic manipulation is key when solving equations involving functions, especially when dealing with composite functions. It includes actions like factoring, simplifying, or rearranging expressions to isolate variables or functions. Let's examine the exercise we solved:

For \( f(g(h(x))) \):

• Start by substituting \( h(x) = \frac{x}{2-x} \) into \( g(x) \), resulting in \( g(h(x)) = \left(\frac{x}{2-x}\right)^2 = \frac{x^2}{(2-x)^2} \)
• Next, substitute \( g(h(x)) \) into \( f(x) \), giving \( f(g(h(x))) = \frac{x^2}{(2-x)^2} + 2 \)

For \( g(h(f(x))) \):

• Substitute \( f(x) = x+2 \) into \( h(x) \), to get \( h(f(x)) = \frac{x+2}{-(x)} \)
• Then substitute the result into \( g(x) \), producing \( g(h(f(x))) = \left(\frac{x+2}{-x}\right)^2 = \frac{(x+2)^2}{x^2} \)

By strategically applying these substitutions, we manipulate and simplify the functions at each step. This process is also fundamental for verifying solutions and ensuring consistency with mathematical rules. Algebraic manipulation not only helps solve composite functions but also deepens comprehension of how different mathematical components come together.